#6868158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,7,8,5)(17,18,22,26,21,23,33,37)(19,27,30,35,28,40,34,24)(20,25,32,39,31,36,44,47)(29,41,43,46,42,48,45,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(18,23)(19,28)(25,36)(26,37)(29,42)(30,34)(39,47)(43,45);(10,14)(15,16)(17,19,21,28)(18,24,23,35)(20,29,31,42)(22,34,33,30)(25,38,36,46)(26,40,37,27)(32,45,44,43)(39,48,47,41);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,31)(18,25,23,36)(19,29,28,42)(22,32,33,44)(24,38,35,46)(26,39,37,47)(27,41,40,48)(30,43,34,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,31)(25,36)(29,42)(32,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775354);